LMFDB - Embedded newform 32.6.b.a.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,6,Mod(17,32)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("32.17");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	32.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.13228223402$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.218489.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 8x + 64 $ x^4 - x^3 - 2x^2 - 8x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.1
Root			$2.38600 - 1.51888i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.6.b.a.17.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-23.6095i q^{3} -1.38521i q^{5} -160.704 q^{7} -314.408 q^{9} +O(q^{10})$	$q-23.6095i q^{3} -1.38521i q^{5} -160.704 q^{7} -314.408 q^{9} -129.129i q^{11} -759.659i q^{13} -32.7041 q^{15} +323.408 q^{17} +198.511i q^{19} +3794.14i q^{21} +1193.15 q^{23} +3123.08 q^{25} +1685.91i q^{27} -5987.24i q^{29} +4872.45 q^{31} -3048.67 q^{33} +222.609i q^{35} +3698.56i q^{37} -17935.2 q^{39} -10437.9 q^{41} -9873.11i q^{43} +435.521i q^{45} +6297.98 q^{47} +9018.79 q^{49} -7635.50i q^{51} -21728.1i q^{53} -178.871 q^{55} +4686.75 q^{57} +33513.4i q^{59} +48506.8i q^{61} +50526.7 q^{63} -1052.29 q^{65} +33182.4i q^{67} -28169.7i q^{69} -59464.1 q^{71} +51278.6 q^{73} -73734.4i q^{75} +20751.6i q^{77} +73724.5 q^{79} -36597.7 q^{81} -61628.0i q^{83} -447.988i q^{85} -141356. q^{87} -106735. q^{89} +122080. i q^{91} -115036. i q^{93} +274.980 q^{95} +12562.9 q^{97} +40599.2i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 96 q^{7} - 164 q^{9}+O(q^{10}) $ 4 * q - 96 * q^7 - 164 * q^9	$ 4 q - 96 q^{7} - 164 q^{9} + 416 q^{15} + 200 q^{17} - 2336 q^{23} + 1556 q^{25} + 12928 q^{31} - 2352 q^{33} - 35104 q^{39} - 4568 q^{41} + 54720 q^{47} + 9828 q^{49} - 85472 q^{55} - 2032 q^{57} + 153440 q^{63} - 19520 q^{65} - 206688 q^{71} + 39976 q^{73} + 247872 q^{79} + 29684 q^{81} - 307872 q^{87} - 84632 q^{89} + 259744 q^{95} - 99576 q^{97}+O(q^{100}) $ 4 * q - 96 * q^7 - 164 * q^9 + 416 * q^15 + 200 * q^17 - 2336 * q^23 + 1556 * q^25 + 12928 * q^31 - 2352 * q^33 - 35104 * q^39 - 4568 * q^41 + 54720 * q^47 + 9828 * q^49 - 85472 * q^55 - 2032 * q^57 + 153440 * q^63 - 19520 * q^65 - 206688 * q^71 + 39976 * q^73 + 247872 * q^79 + 29684 * q^81 - 307872 * q^87 - 84632 * q^89 + 259744 * q^95 - 99576 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$.

$n$	$5$	$31$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 23.6095i	− 1.51455i	−0.653096	−	0.757275i	$-0.726530\pi$
$3$	− 23.6095i	− 1.51455i	0.653096	−	0.757275i	$-0.273470\pi$
$4$	0	0
$5$	− 1.38521i	− 0.0247794i	−0.999923	−	0.0123897i	$-0.996056\pi$
$5$	− 1.38521i	− 0.0247794i	0.999923	−	0.0123897i	$-0.00394386\pi$
$6$	0	0
$7$	−160.704	−1.23960	−0.619800	−	0.784760i	$-0.712786\pi$
$7$	−160.704	−1.23960	−0.619800	+	0.784760i	$0.712786\pi$
$8$	0	0
$9$	−314.408	−1.29386
$10$	0	0
$11$	− 129.129i	− 0.321768i	−0.986973	−	0.160884i	$-0.948566\pi$
$11$	− 129.129i	− 0.321768i	0.986973	−	0.160884i	$-0.0514345\pi$
$12$	0	0
$13$	− 759.659i	− 1.24670i	−0.781945	−	0.623348i	$-0.785772\pi$
$13$	− 759.659i	− 1.24670i	0.781945	−	0.623348i	$-0.214228\pi$
$14$	0	0
$15$	−32.7041	−0.0375296
$16$	0	0
$17$	323.408	0.271412	0.135706	−	0.990749i	$-0.456670\pi$
$17$	323.408	0.271412	0.135706	+	0.990749i	$0.456670\pi$
$18$	0	0
$19$	198.511i	0.126154i	0.998009	+	0.0630771i	$0.0200914\pi$
$19$	198.511i	0.126154i	−0.998009	+	0.0630771i	$0.979909\pi$
$20$	0	0
$21$	3794.14i	1.87744i
$22$	0	0
$23$	1193.15	0.470302	0.235151	−	0.971959i	$-0.424442\pi$
$23$	1193.15	0.470302	0.235151	+	0.971959i	$0.424442\pi$
$24$	0	0
$25$	3123.08	0.999386
$26$	0	0
$27$	1685.91i	0.445066i
$28$	0	0
$29$	− 5987.24i	− 1.32200i	−0.750386	−	0.661000i	$-0.770132\pi$
$29$	− 5987.24i	− 1.32200i	0.750386	−	0.661000i	$-0.229868\pi$
$30$	0	0
$31$	4872.45	0.910632	0.455316	−	0.890330i	$-0.349526\pi$
$31$	4872.45	0.910632	0.455316	+	0.890330i	$0.349526\pi$
$32$	0	0
$33$	−3048.67	−0.487333
$34$	0	0
$35$	222.609i	0.0307165i
$36$	0	0
$37$	3698.56i	0.444149i	0.975030	+	0.222074i	$0.0712828\pi$
$37$	3698.56i	0.444149i	−0.975030	+	0.222074i	$0.928717\pi$
$38$	0	0
$39$	−17935.2	−1.88818
$40$	0	0
$41$	−10437.9	−0.969734	−0.484867	−	0.874588i	$-0.661132\pi$
$41$	−10437.9	−0.969734	−0.484867	+	0.874588i	$0.661132\pi$
$42$	0	0
$43$	− 9873.11i	− 0.814297i	−0.913362	−	0.407148i	$-0.866523\pi$
$43$	− 9873.11i	− 0.814297i	0.913362	−	0.407148i	$-0.133477\pi$
$44$	0	0
$45$	435.521i	0.0320610i
$46$	0	0
$47$	6297.98	0.415869	0.207935	−	0.978143i	$-0.433326\pi$
$47$	6297.98	0.415869	0.207935	+	0.978143i	$0.433326\pi$
$48$	0	0
$49$	9018.79	0.536609
$50$	0	0
$51$	− 7635.50i	− 0.411067i
$52$	0	0
$53$	− 21728.1i	− 1.06251i	−0.847212	−	0.531255i	$-0.821721\pi$
$53$	− 21728.1i	− 1.06251i	0.847212	−	0.531255i	$-0.178279\pi$
$54$	0	0
$55$	−178.871	−0.00797320
$56$	0	0
$57$	4686.75	0.191067
$58$	0	0
$59$	33513.4i	1.25340i	0.779261	+	0.626699i	$0.215594\pi$
$59$	33513.4i	1.25340i	−0.779261	+	0.626699i	$0.784406\pi$
$60$	0	0
$61$	48506.8i	1.66908i	0.550944	+	0.834542i	$0.314268\pi$
$61$	48506.8i	1.66908i	−0.550944	+	0.834542i	$0.685732\pi$
$62$	0	0
$63$	50526.7	1.60387
$64$	0	0
$65$	−1052.29	−0.0308923
$66$	0	0
$67$	33182.4i	0.903068i	0.892254	+	0.451534i	$0.149123\pi$
$67$	33182.4i	0.903068i	−0.892254	+	0.451534i	$0.850877\pi$
$68$	0	0
$69$	− 28169.7i	− 0.712295i
$70$	0	0
$71$	−59464.1	−1.39994	−0.699970	−	0.714173i	$-0.746803\pi$
$71$	−59464.1	−1.39994	−0.699970	+	0.714173i	$0.746803\pi$
$72$	0	0
$73$	51278.6	1.12624	0.563118	−	0.826377i	$-0.309602\pi$
$73$	51278.6	1.12624	0.563118	+	0.826377i	$0.309602\pi$
$74$	0	0
$75$	− 73734.4i	− 1.51362i
$76$	0	0
$77$	20751.6i	0.398863i
$78$	0	0
$79$	73724.5	1.32906	0.664530	−	0.747262i	$-0.268632\pi$
$79$	73724.5	1.32906	0.664530	+	0.747262i	$0.268632\pi$
$80$	0	0
$81$	−36597.7	−0.619785
$82$	0	0
$83$	− 61628.0i	− 0.981935i	−0.871178	−	0.490967i	$-0.836643\pi$
$83$	− 61628.0i	− 0.981935i	0.871178	−	0.490967i	$-0.163357\pi$
$84$	0	0
$85$	− 447.988i	− 0.00672541i
$86$	0	0
$87$	−141356.	−2.00223
$88$	0	0
$89$	−106735.	−1.42834	−0.714169	−	0.699974i	$-0.753195\pi$
$89$	−106735.	−1.42834	−0.714169	+	0.699974i	$0.753195\pi$
$90$	0	0
$91$	122080.i	1.54540i
$92$	0	0
$93$	− 115036.i	− 1.37920i
$94$	0	0
$95$	274.980	0.00312602
$96$	0	0
$97$	12562.9	0.135569	0.0677846	−	0.997700i	$-0.478407\pi$
$97$	12562.9	0.135569	0.0677846	+	0.997700i	$0.478407\pi$
$98$	0	0
$99$	40599.2i	0.416323i
$100$	0	0
$101$	64962.8i	0.633667i	0.948481	+	0.316834i	$0.102620\pi$
$101$	64962.8i	0.633667i	−0.948481	+	0.316834i	$0.897380\pi$
$102$	0	0
$103$	69035.9	0.641183	0.320591	−	0.947218i	$-0.396118\pi$
$103$	69035.9	0.641183	0.320591	+	0.947218i	$0.396118\pi$
$104$	0	0
$105$	5255.68	0.0465217
$106$	0	0
$107$	− 187322.i	− 1.58172i	−0.612000	−	0.790858i	$-0.709635\pi$
$107$	− 187322.i	− 1.58172i	0.612000	−	0.790858i	$-0.290365\pi$
$108$	0	0
$109$	− 49350.5i	− 0.397855i	−0.980014	−	0.198928i	$-0.936254\pi$
$109$	− 49350.5i	− 0.397855i	0.980014	−	0.198928i	$-0.0637459\pi$
$110$	0	0
$111$	87321.2	0.672686
$112$	0	0
$113$	171100.	1.26053	0.630266	−	0.776379i	$-0.282946\pi$
$113$	171100.	1.26053	0.630266	+	0.776379i	$0.282946\pi$
$114$	0	0
$115$	− 1652.76i	− 0.0116538i
$116$	0	0
$117$	238843.i	1.61305i
$118$	0	0
$119$	−51973.0	−0.336442
$120$	0	0
$121$	144377.	0.896466
$122$	0	0
$123$	246433.i	1.46871i
$124$	0	0
$125$	− 8654.89i	− 0.0495435i
$126$	0	0
$127$	90479.7	0.497785	0.248892	−	0.968531i	$-0.419933\pi$
$127$	90479.7	0.497785	0.248892	+	0.968531i	$0.419933\pi$
$128$	0	0
$129$	−233099.	−1.23329
$130$	0	0
$131$	94491.0i	0.481074i	0.970640	+	0.240537i	$0.0773236\pi$
$131$	94491.0i	0.481074i	−0.970640	+	0.240537i	$0.922676\pi$
$132$	0	0
$133$	− 31901.6i	− 0.156381i
$134$	0	0
$135$	2335.34	0.0110285
$136$	0	0
$137$	−5516.81	−0.0251123	−0.0125561	−	0.999921i	$-0.503997\pi$
$137$	−5516.81	−0.0251123	−0.0125561	+	0.999921i	$0.503997\pi$
$138$	0	0
$139$	− 164489.i	− 0.722104i	−0.932546	−	0.361052i	$-0.882418\pi$
$139$	− 164489.i	− 0.722104i	0.932546	−	0.361052i	$-0.117582\pi$
$140$	0	0
$141$	− 148692.i	− 0.629854i
$142$	0	0
$143$	−98094.2	−0.401147
$144$	0	0
$145$	−8293.57	−0.0327583
$146$	0	0
$147$	− 212929.i	− 0.812722i
$148$	0	0
$149$	− 423122.i	− 1.56135i	−0.624939	−	0.780674i	$-0.714876\pi$
$149$	− 423122.i	− 1.56135i	0.624939	−	0.780674i	$-0.285124\pi$
$150$	0	0
$151$	39975.1	0.142675	0.0713373	−	0.997452i	$-0.477273\pi$
$151$	39975.1	0.142675	0.0713373	+	0.997452i	$0.477273\pi$
$152$	0	0
$153$	−101682.	−0.351169
$154$	0	0
$155$	− 6749.36i	− 0.0225649i
$156$	0	0
$157$	367453.i	1.18974i	0.803822	+	0.594870i	$0.202797\pi$
$157$	367453.i	1.18974i	−0.803822	+	0.594870i	$0.797203\pi$
$158$	0	0
$159$	−512991.	−1.60922
$160$	0	0
$161$	−191744.	−0.582986
$162$	0	0
$163$	− 16030.9i	− 0.0472596i	−0.999721	−	0.0236298i	$-0.992478\pi$
$163$	− 16030.9i	− 0.0472596i	0.999721	−	0.0236298i	$-0.00752230\pi$
$164$	0	0
$165$	4223.05i	0.0120758i
$166$	0	0
$167$	248107.	0.688412	0.344206	−	0.938894i	$-0.388148\pi$
$167$	248107.	0.688412	0.344206	+	0.938894i	$0.388148\pi$
$168$	0	0
$169$	−205789.	−0.554251
$170$	0	0
$171$	− 62413.6i	− 0.163226i
$172$	0	0
$173$	574094.i	1.45837i	0.684317	+	0.729185i	$0.260101\pi$
$173$	574094.i	1.45837i	−0.684317	+	0.729185i	$0.739899\pi$
$174$	0	0
$175$	−501892.	−1.23884
$176$	0	0
$177$	791235.	1.89833
$178$	0	0
$179$	305296.i	0.712177i	0.934452	+	0.356088i	$0.115890\pi$
$179$	305296.i	0.712177i	−0.934452	+	0.356088i	$0.884110\pi$
$180$	0	0
$181$	− 421682.i	− 0.956728i	−0.878162	−	0.478364i	$-0.841230\pi$
$181$	− 421682.i	− 0.956728i	0.878162	−	0.478364i	$-0.158770\pi$
$182$	0	0
$183$	1.14522e6	2.52791
$184$	0	0
$185$	5123.28	0.0110057
$186$	0	0
$187$	− 41761.4i	− 0.0873315i
$188$	0	0
$189$	− 270932.i	− 0.551705i
$190$	0	0
$191$	424231.	0.841431	0.420716	−	0.907193i	$-0.361779\pi$
$191$	424231.	0.841431	0.420716	+	0.907193i	$0.361779\pi$
$192$	0	0
$193$	−373902.	−0.722545	−0.361272	−	0.932460i	$-0.617658\pi$
$193$	−373902.	−0.722545	−0.361272	+	0.932460i	$0.617658\pi$
$194$	0	0
$195$	24843.9i	0.0467880i
$196$	0	0
$197$	− 112848.i	− 0.207171i	−0.994621	−	0.103586i	$-0.966968\pi$
$197$	− 112848.i	− 0.207171i	0.994621	−	0.103586i	$-0.0330316\pi$
$198$	0	0
$199$	−262938.	−0.470675	−0.235338	−	0.971914i	$-0.575620\pi$
$199$	−262938.	−0.470675	−0.235338	+	0.971914i	$0.575620\pi$
$200$	0	0
$201$	783419.	1.36774
$202$	0	0
$203$	962173.i	1.63875i
$204$	0	0
$205$	14458.6i	0.0240294i
$206$	0	0
$207$	−375137.	−0.608505
$208$	0	0
$209$	25633.6	0.0405923
$210$	0	0
$211$	− 272968.i	− 0.422090i	−0.977476	−	0.211045i	$-0.932313\pi$
$211$	− 272968.i	− 0.422090i	0.977476	−	0.211045i	$-0.0676866\pi$
$212$	0	0
$213$	1.40392e6i	2.12028i
$214$	0	0
$215$	−13676.3	−0.0201777
$216$	0	0
$217$	−783022.	−1.12882
$218$	0	0
$219$	− 1.21066e6i	− 1.70574i
$220$	0	0
$221$	− 245680.i	− 0.338368i
$222$	0	0
$223$	−1.00553e6	−1.35405	−0.677023	−	0.735962i	$-0.736730\pi$
$223$	−1.00553e6	−1.35405	−0.677023	+	0.735962i	$0.736730\pi$
$224$	0	0
$225$	−981922.	−1.29307
$226$	0	0
$227$	− 554991.i	− 0.714861i	−0.933940	−	0.357430i	$-0.883653\pi$
$227$	− 554991.i	− 0.714861i	0.933940	−	0.357430i	$-0.116347\pi$
$228$	0	0
$229$	− 476013.i	− 0.599832i	−0.953966	−	0.299916i	$-0.903041\pi$
$229$	− 476013.i	− 0.599832i	0.953966	−	0.299916i	$-0.0969587\pi$
$230$	0	0
$231$	489934.	0.604098
$232$	0	0
$233$	914141.	1.10312	0.551561	−	0.834135i	$-0.314032\pi$
$233$	914141.	1.10312	0.551561	+	0.834135i	$0.314032\pi$
$234$	0	0
$235$	− 8724.01i	− 0.0103050i
$236$	0	0
$237$	− 1.74060e6i	− 2.01293i
$238$	0	0
$239$	−375827.	−0.425592	−0.212796	−	0.977097i	$-0.568257\pi$
$239$	−375827.	−0.425592	−0.212796	+	0.977097i	$0.568257\pi$
$240$	0	0
$241$	612110.	0.678870	0.339435	−	0.940629i	$-0.389764\pi$
$241$	612110.	0.678870	0.339435	+	0.940629i	$0.389764\pi$
$242$	0	0
$243$	1.27373e6i	1.38376i
$244$	0	0
$245$	− 12492.9i	− 0.0132968i
$246$	0	0
$247$	150801.	0.157276
$248$	0	0
$249$	−1.45501e6	−1.48719
$250$	0	0
$251$	− 462623.i	− 0.463493i	−0.972776	−	0.231746i	$-0.925556\pi$
$251$	− 462623.i	− 0.463493i	0.972776	−	0.231746i	$-0.0744440\pi$
$252$	0	0
$253$	− 154071.i	− 0.151328i
$254$	0	0
$255$	−10576.8	−0.0101860
$256$	0	0
$257$	−583345.	−0.550925	−0.275463	−	0.961312i	$-0.588831\pi$
$257$	−583345.	−0.550925	−0.275463	+	0.961312i	$0.588831\pi$
$258$	0	0
$259$	− 594374.i	− 0.550567i
$260$	0	0
$261$	1.88244e6i	1.71048i
$262$	0	0
$263$	411975.	0.367267	0.183633	−	0.982995i	$-0.441214\pi$
$263$	411975.	0.367267	0.183633	+	0.982995i	$0.441214\pi$
$264$	0	0
$265$	−30098.0	−0.0263283
$266$	0	0
$267$	2.51995e6i	2.16329i
$268$	0	0
$269$	1.14460e6i	0.964436i	0.876051	+	0.482218i	$0.160169\pi$
$269$	1.14460e6i	0.964436i	−0.876051	+	0.482218i	$0.839831\pi$
$270$	0	0
$271$	1.21607e6	1.00586	0.502928	−	0.864329i	$-0.332256\pi$
$271$	1.21607e6	1.00586	0.502928	+	0.864329i	$0.332256\pi$
$272$	0	0
$273$	2.88225e6	2.34059
$274$	0	0
$275$	− 403281.i	− 0.321570i
$276$	0	0
$277$	− 806054.i	− 0.631196i	−0.948893	−	0.315598i	$-0.897795\pi$
$277$	− 806054.i	− 0.631196i	0.948893	−	0.315598i	$-0.102205\pi$
$278$	0	0
$279$	−1.53194e6	−1.17823
$280$	0	0
$281$	1.19824e6	0.905272	0.452636	−	0.891695i	$-0.350484\pi$
$281$	1.19824e6	0.905272	0.452636	+	0.891695i	$0.350484\pi$
$282$	0	0
$283$	− 1.46287e6i	− 1.08578i	−0.839805	−	0.542888i	$-0.817331\pi$
$283$	− 1.46287e6i	− 1.08578i	0.839805	−	0.542888i	$-0.182669\pi$
$284$	0	0
$285$	− 6492.13i	− 0.00473451i
$286$	0	0
$287$	1.67741e6	1.20208
$288$	0	0
$289$	−1.31526e6	−0.926336
$290$	0	0
$291$	− 296604.i	− 0.205326i
$292$	0	0
$293$	− 750723.i	− 0.510870i	−0.966826	−	0.255435i	$-0.917781\pi$
$293$	− 750723.i	− 0.510870i	0.966826	−	0.255435i	$-0.0822187\pi$
$294$	0	0
$295$	46423.1	0.0310584
$296$	0	0
$297$	217700.	0.143208
$298$	0	0
$299$	− 906390.i	− 0.586323i
$300$	0	0
$301$	1.58665e6i	1.00940i
$302$	0	0
$303$	1.53374e6	0.959721
$304$	0	0
$305$	67192.0	0.0413588
$306$	0	0
$307$	2.06754e6i	1.25201i	0.779819	+	0.626005i	$0.215311\pi$
$307$	2.06754e6i	1.25201i	−0.779819	+	0.626005i	$0.784689\pi$
$308$	0	0
$309$	− 1.62990e6i	− 0.971103i
$310$	0	0
$311$	−3.06896e6	−1.79924	−0.899621	−	0.436671i	$-0.856157\pi$
$311$	−3.06896e6	−1.79924	−0.899621	+	0.436671i	$0.856157\pi$
$312$	0	0
$313$	−115148.	−0.0664345	−0.0332173	−	0.999448i	$-0.510575\pi$
$313$	−115148.	−0.0664345	−0.0332173	+	0.999448i	$0.510575\pi$
$314$	0	0
$315$	− 69989.9i	− 0.0397429i
$316$	0	0
$317$	1.29930e6i	0.726210i	0.931748	+	0.363105i	$0.118283\pi$
$317$	1.29930e6i	0.726210i	−0.931748	+	0.363105i	$0.881717\pi$
$318$	0	0
$319$	−773127.	−0.425377
$320$	0	0
$321$	−4.42257e6	−2.39559
$322$	0	0
$323$	64200.2i	0.0342397i
$324$	0	0
$325$	− 2.37248e6i	− 1.24593i
$326$	0	0
$327$	−1.16514e6	−0.602571
$328$	0	0
$329$	−1.01211e6	−0.515512
$330$	0	0
$331$	2.02113e6i	1.01397i	0.861955	+	0.506985i	$0.169240\pi$
$331$	2.02113e6i	1.01397i	−0.861955	+	0.506985i	$0.830760\pi$
$332$	0	0
$333$	− 1.16286e6i	− 0.574667i
$334$	0	0
$335$	45964.5	0.0223775
$336$	0	0
$337$	2.88553e6	1.38405	0.692023	−	0.721875i	$-0.256719\pi$
$337$	2.88553e6	1.38405	0.692023	+	0.721875i	$0.256719\pi$
$338$	0	0
$339$	− 4.03958e6i	− 1.90914i
$340$	0	0
$341$	− 629175.i	− 0.293012i
$342$	0	0
$343$	1.25160e6	0.574419
$344$	0	0
$345$	−39020.9	−0.0176502
$346$	0	0
$347$	− 1.01894e6i	− 0.454281i	−0.973862	−	0.227141i	$-0.927062\pi$
$347$	− 1.01894e6i	− 0.454281i	0.973862	−	0.227141i	$-0.0729377\pi$
$348$	0	0
$349$	1.53786e6i	0.675854i	0.941172	+	0.337927i	$0.109726\pi$
$349$	1.53786e6i	0.675854i	−0.941172	+	0.337927i	$0.890274\pi$
$350$	0	0
$351$	1.28072e6	0.554862
$352$	0	0
$353$	−490388.	−0.209461	−0.104731	−	0.994501i	$-0.533398\pi$
$353$	−490388.	−0.209461	−0.104731	+	0.994501i	$0.533398\pi$
$354$	0	0
$355$	82370.2i	0.0346896i
$356$	0	0
$357$	1.22706e6i	0.509558i
$358$	0	0
$359$	−3.19930e6	−1.31014	−0.655072	−	0.755567i	$-0.727362\pi$
$359$	−3.19930e6	−1.31014	−0.655072	+	0.755567i	$0.727362\pi$
$360$	0	0
$361$	2.43669e6	0.984085
$362$	0	0
$363$	− 3.40866e6i	− 1.35774i
$364$	0	0
$365$	− 71031.6i	− 0.0279074i
$366$	0	0
$367$	2.06745e6	0.801252	0.400626	−	0.916242i	$-0.368793\pi$
$367$	2.06745e6	0.801252	0.400626	+	0.916242i	$0.368793\pi$
$368$	0	0
$369$	3.28175e6	1.25470
$370$	0	0
$371$	3.49180e6i	1.31709i
$372$	0	0
$373$	4.93913e6i	1.83814i	0.394095	+	0.919070i	$0.371058\pi$
$373$	4.93913e6i	1.83814i	−0.394095	+	0.919070i	$0.628942\pi$
$374$	0	0
$375$	−204338.	−0.0750361
$376$	0	0
$377$	−4.54826e6	−1.64813
$378$	0	0
$379$	5.21670e6i	1.86551i	0.360510	+	0.932755i	$0.382603\pi$
$379$	5.21670e6i	1.86551i	−0.360510	+	0.932755i	$0.617397\pi$
$380$	0	0
$381$	− 2.13618e6i	− 0.753920i
$382$	0	0
$383$	4.22327e6	1.47113	0.735566	−	0.677453i	$-0.236916\pi$
$383$	4.22327e6	1.47113	0.735566	+	0.677453i	$0.236916\pi$
$384$	0	0
$385$	28745.2	0.00988358
$386$	0	0
$387$	3.10418e6i	1.05359i
$388$	0	0
$389$	− 615402.i	− 0.206198i	−0.994671	−	0.103099i	$-0.967124\pi$
$389$	− 615402.i	− 0.206198i	0.994671	−	0.103099i	$-0.0328759\pi$
$390$	0	0
$391$	385875.	0.127645
$392$	0	0
$393$	2.23088e6	0.728611
$394$	0	0
$395$	− 102124.i	− 0.0329332i
$396$	0	0
$397$	− 1.60554e6i	− 0.511263i	−0.966774	−	0.255632i	$-0.917717\pi$
$397$	− 1.60554e6i	− 0.511263i	0.966774	−	0.255632i	$-0.0822835\pi$
$398$	0	0
$399$	−753180.	−0.236846
$400$	0	0
$401$	1.47973e6	0.459539	0.229769	−	0.973245i	$-0.426203\pi$
$401$	1.47973e6	0.459539	0.229769	+	0.973245i	$0.426203\pi$
$402$	0	0
$403$	− 3.70140e6i	− 1.13528i
$404$	0	0
$405$	50695.4i	0.0153579i
$406$	0	0
$407$	477592.	0.142913
$408$	0	0
$409$	1.15560e6	0.341584	0.170792	−	0.985307i	$-0.445367\pi$
$409$	1.15560e6	0.341584	0.170792	+	0.985307i	$0.445367\pi$
$410$	0	0
$411$	130249.i	0.0380338i
$412$	0	0
$413$	− 5.38575e6i	− 1.55371i
$414$	0	0
$415$	−85367.6	−0.0243317
$416$	0	0
$417$	−3.88350e6	−1.09366
$418$	0	0
$419$	− 1.84397e6i	− 0.513121i	−0.966528	−	0.256560i	$-0.917411\pi$
$419$	− 1.84397e6i	− 0.513121i	0.966528	−	0.256560i	$-0.0825893\pi$
$420$	0	0
$421$	4.13061e6i	1.13582i	0.823091	+	0.567909i	$0.192248\pi$
$421$	4.13061e6i	1.13582i	−0.823091	+	0.567909i	$0.807752\pi$
$422$	0	0
$423$	−1.98014e6	−0.538077
$424$	0	0
$425$	1.01003e6	0.271245
$426$	0	0
$427$	− 7.79524e6i	− 2.06900i
$428$	0	0
$429$	2.31595e6i	0.607556i
$430$	0	0
$431$	−3.44366e6	−0.892950	−0.446475	−	0.894796i	$-0.647321\pi$
$431$	−3.44366e6	−0.892950	−0.446475	+	0.894796i	$0.647321\pi$
$432$	0	0
$433$	−2.41696e6	−0.619511	−0.309755	−	0.950816i	$-0.600247\pi$
$433$	−2.41696e6	−0.619511	−0.309755	+	0.950816i	$0.600247\pi$
$434$	0	0
$435$	195807.i	0.0496141i
$436$	0	0
$437$	236854.i	0.0593305i
$438$	0	0
$439$	3.22639e6	0.799015	0.399507	−	0.916730i	$-0.369181\pi$
$439$	3.22639e6	0.799015	0.399507	+	0.916730i	$0.369181\pi$
$440$	0	0
$441$	−2.83558e6	−0.694298
$442$	0	0
$443$	6.44624e6i	1.56062i	0.625393	+	0.780310i	$0.284939\pi$
$443$	6.44624e6i	1.56062i	−0.625393	+	0.780310i	$0.715061\pi$
$444$	0	0
$445$	147850.i	0.0353933i
$446$	0	0
$447$	−9.98969e6	−2.36474
$448$	0	0
$449$	−1.82600e6	−0.427450	−0.213725	−	0.976894i	$-0.568560\pi$
$449$	−1.82600e6	−0.427450	−0.213725	+	0.976894i	$0.568560\pi$
$450$	0	0
$451$	1.34783e6i	0.312029i
$452$	0	0
$453$	− 943791.i	− 0.216088i
$454$	0	0
$455$	169107.	0.0382941
$456$	0	0
$457$	2.26865e6	0.508132	0.254066	−	0.967187i	$-0.418232\pi$
$457$	2.26865e6	0.508132	0.254066	+	0.967187i	$0.418232\pi$
$458$	0	0
$459$	545237.i	0.120796i
$460$	0	0
$461$	− 2.82378e6i	− 0.618840i	−0.950925	−	0.309420i	$-0.899865\pi$
$461$	− 2.82378e6i	− 0.618840i	0.950925	−	0.309420i	$-0.100135\pi$
$462$	0	0
$463$	−3.05836e6	−0.663035	−0.331517	−	0.943449i	$-0.607561\pi$
$463$	−3.05836e6	−0.663035	−0.331517	+	0.943449i	$0.607561\pi$
$464$	0	0
$465$	−159349.	−0.0341756
$466$	0	0
$467$	− 5.19478e6i	− 1.10224i	−0.834427	−	0.551119i	$-0.814201\pi$
$467$	− 5.19478e6i	− 1.10224i	0.834427	−	0.551119i	$-0.185799\pi$
$468$	0	0
$469$	− 5.33254e6i	− 1.11944i
$470$	0	0
$471$	8.67537e6	1.80192
$472$	0	0
$473$	−1.27491e6	−0.262014
$474$	0	0
$475$	619967.i	0.126077i
$476$	0	0
$477$	6.83151e6i	1.37474i
$478$	0	0
$479$	8.27094e6	1.64708	0.823542	−	0.567255i	$-0.191994\pi$
$479$	8.27094e6	1.64708	0.823542	+	0.567255i	$0.191994\pi$
$480$	0	0
$481$	2.80965e6	0.553719
$482$	0	0
$483$	4.52699e6i	0.882961i
$484$	0	0
$485$	− 17402.2i	− 0.00335932i
$486$	0	0
$487$	4.95663e6	0.947031	0.473515	−	0.880786i	$-0.342985\pi$
$487$	4.95663e6	0.947031	0.473515	+	0.880786i	$0.342985\pi$
$488$	0	0
$489$	−378482.	−0.0715770
$490$	0	0
$491$	− 6.01801e6i	− 1.12655i	−0.826271	−	0.563273i	$-0.809542\pi$
$491$	− 6.01801e6i	− 1.12655i	0.826271	−	0.563273i	$-0.190458\pi$
$492$	0	0
$493$	− 1.93632e6i	− 0.358806i
$494$	0	0
$495$	56238.4	0.0103162
$496$	0	0
$497$	9.55613e6	1.73537
$498$	0	0
$499$	7.87834e6i	1.41639i	0.706016	+	0.708196i	$0.250491\pi$
$499$	7.87834e6i	1.41639i	−0.706016	+	0.708196i	$0.749509\pi$
$500$	0	0
$501$	− 5.85769e6i	− 1.04263i
$502$	0	0
$503$	−9.09472e6	−1.60276	−0.801382	−	0.598153i	$-0.795901\pi$
$503$	−9.09472e6	−1.60276	−0.801382	+	0.598153i	$0.795901\pi$
$504$	0	0
$505$	89987.0	0.0157019
$506$	0	0
$507$	4.85858e6i	0.839440i
$508$	0	0
$509$	1.07691e7i	1.84241i	0.389076	+	0.921206i	$0.372794\pi$
$509$	1.07691e7i	1.84241i	−0.389076	+	0.921206i	$0.627206\pi$
$510$	0	0
$511$	−8.24068e6	−1.39608
$512$	0	0
$513$	−334672.	−0.0561470
$514$	0	0
$515$	− 95629.1i	− 0.0158881i
$516$	0	0
$517$	− 813253.i	− 0.133813i
$518$	0	0
$519$	1.35541e7	2.20877
$520$	0	0
$521$	−1.88429e6	−0.304126	−0.152063	−	0.988371i	$-0.548592\pi$
$521$	−1.88429e6	−0.304126	−0.152063	+	0.988371i	$0.548592\pi$
$522$	0	0
$523$	2.14270e6i	0.342537i	0.985224	+	0.171269i	$0.0547866\pi$
$523$	2.14270e6i	0.342537i	−0.985224	+	0.171269i	$0.945213\pi$
$524$	0	0
$525$	1.18494e7i	1.87628i
$526$	0	0
$527$	1.57579e6	0.247156
$528$	0	0
$529$	−5.01273e6	−0.778816
$530$	0	0
$531$	− 1.05369e7i	− 1.62172i
$532$	0	0
$533$	7.92923e6i	1.20896i
$534$	0	0
$535$	−259479.	−0.0391939
$536$	0	0
$537$	7.20787e6	1.07863
$538$	0	0
$539$	− 1.16459e6i	− 0.172664i
$540$	0	0
$541$	− 9.23309e6i	− 1.35629i	−0.734926	−	0.678147i	$-0.762783\pi$
$541$	− 9.23309e6i	− 1.35629i	0.734926	−	0.678147i	$-0.237217\pi$
$542$	0	0
$543$	−9.95569e6	−1.44901
$544$	0	0
$545$	−68360.7	−0.00985859
$546$	0	0
$547$	− 6.30413e6i	− 0.900858i	−0.892812	−	0.450429i	$-0.851271\pi$
$547$	− 6.30413e6i	− 0.900858i	0.892812	−	0.450429i	$-0.148729\pi$
$548$	0	0
$549$	− 1.52509e7i	− 2.15956i
$550$	0	0
$551$	1.18853e6	0.166776
$552$	0	0
$553$	−1.18478e7	−1.64750
$554$	0	0
$555$	− 120958.i	− 0.0166687i
$556$	0	0
$557$	− 6.50282e6i	− 0.888104i	−0.896001	−	0.444052i	$-0.853540\pi$
$557$	− 6.50282e6i	− 0.888104i	0.896001	−	0.444052i	$-0.146460\pi$
$558$	0	0
$559$	−7.50020e6	−1.01518
$560$	0	0
$561$	−985966.	−0.132268
$562$	0	0
$563$	6.06434e6i	0.806329i	0.915127	+	0.403165i	$0.132090\pi$
$563$	6.06434e6i	0.806329i	−0.915127	+	0.403165i	$0.867910\pi$
$564$	0	0
$565$	− 237009.i	− 0.0312352i
$566$	0	0
$567$	5.88140e6	0.768286
$568$	0	0
$569$	8.67931e6	1.12384	0.561920	−	0.827191i	$-0.310063\pi$
$569$	8.67931e6	1.12384	0.561920	+	0.827191i	$0.310063\pi$
$570$	0	0
$571$	− 5.13091e6i	− 0.658573i	−0.944230	−	0.329287i	$-0.893192\pi$
$571$	− 5.13091e6i	− 0.658573i	0.944230	−	0.329287i	$-0.106808\pi$
$572$	0	0
$573$	− 1.00159e7i	− 1.27439i
$574$	0	0
$575$	3.72631e6	0.470013
$576$	0	0
$577$	−1.05397e7	−1.31792	−0.658962	−	0.752176i	$-0.729004\pi$
$577$	−1.05397e7	−1.31792	−0.658962	+	0.752176i	$0.729004\pi$
$578$	0	0
$579$	8.82764e6i	1.09433i
$580$	0	0
$581$	9.90387e6i	1.21721i
$582$	0	0
$583$	−2.80574e6	−0.341881
$584$	0	0
$585$	330847.	0.0399704
$586$	0	0
$587$	5.07345e6i	0.607727i	0.952716	+	0.303863i	$0.0982767\pi$
$587$	5.07345e6i	0.607727i	−0.952716	+	0.303863i	$0.901723\pi$
$588$	0	0
$589$	967237.i	0.114880i
$590$	0	0
$591$	−2.66429e6	−0.313771
$592$	0	0
$593$	−1.41356e7	−1.65073	−0.825366	−	0.564599i	$-0.809031\pi$
$593$	−1.41356e7	−1.65073	−0.825366	+	0.564599i	$0.809031\pi$
$594$	0	0
$595$	71993.4i	0.00833682i
$596$	0	0
$597$	6.20784e6i	0.712861i
$598$	0	0
$599$	−7.17896e6	−0.817512	−0.408756	−	0.912644i	$-0.634037\pi$
$599$	−7.17896e6	−0.817512	−0.408756	+	0.912644i	$0.634037\pi$
$600$	0	0
$601$	−3.61001e6	−0.407683	−0.203841	−	0.979004i	$-0.565343\pi$
$601$	−3.61001e6	−0.407683	−0.203841	+	0.979004i	$0.565343\pi$
$602$	0	0
$603$	− 1.04328e7i	− 1.16844i
$604$	0	0
$605$	− 199992.i	− 0.0222138i
$606$	0	0
$607$	−4.03263e6	−0.444239	−0.222119	−	0.975020i	$-0.571297\pi$
$607$	−4.03263e6	−0.444239	−0.222119	+	0.975020i	$0.571297\pi$
$608$	0	0
$609$	2.27164e7	2.48197
$610$	0	0
$611$	− 4.78432e6i	− 0.518462i
$612$	0	0
$613$	1.31815e7i	1.41682i	0.705801	+	0.708410i	$0.250587\pi$
$613$	1.31815e7i	1.41682i	−0.705801	+	0.708410i	$0.749413\pi$
$614$	0	0
$615$	341361.	0.0363937
$616$	0	0
$617$	7.04507e6	0.745027	0.372514	−	0.928027i	$-0.378496\pi$
$617$	7.04507e6	0.745027	0.372514	+	0.928027i	$0.378496\pi$
$618$	0	0
$619$	6.48539e6i	0.680314i	0.940369	+	0.340157i	$0.110480\pi$
$619$	6.48539e6i	0.680314i	−0.940369	+	0.340157i	$0.889520\pi$
$620$	0	0
$621$	2.01155e6i	0.209315i
$622$	0	0
$623$	1.71527e7	1.77057
$624$	0	0
$625$	9.74764e6	0.998158
$626$	0	0
$627$	− 605196.i	− 0.0614791i
$628$	0	0
$629$	1.19614e6i	0.120547i
$630$	0	0
$631$	4.09103e6	0.409034	0.204517	−	0.978863i	$-0.434438\pi$
$631$	4.09103e6	0.409034	0.204517	+	0.978863i	$0.434438\pi$
$632$	0	0
$633$	−6.44463e6	−0.639276
$634$	0	0
$635$	− 125333.i	− 0.0123348i
$636$	0	0
$637$	− 6.85121e6i	− 0.668989i
$638$	0	0
$639$	1.86960e7	1.81133
$640$	0	0
$641$	8.17877e6	0.786218	0.393109	−	0.919492i	$-0.371400\pi$
$641$	8.17877e6	0.786218	0.393109	+	0.919492i	$0.371400\pi$
$642$	0	0
$643$	− 1.41619e7i	− 1.35081i	−0.737448	−	0.675404i	$-0.763969\pi$
$643$	− 1.41619e7i	− 1.35081i	0.737448	−	0.675404i	$-0.236031\pi$
$644$	0	0
$645$	322891.i	0.0305602i
$646$	0	0
$647$	−1.23662e7	−1.16138	−0.580690	−	0.814124i	$-0.697217\pi$
$647$	−1.23662e7	−1.16138	−0.580690	+	0.814124i	$0.697217\pi$
$648$	0	0
$649$	4.32756e6	0.403303
$650$	0	0
$651$	1.84868e7i	1.70965i
$652$	0	0
$653$	− 1.99967e7i	− 1.83517i	−0.397543	−	0.917584i	$-0.630137\pi$
$653$	− 1.99967e7i	− 1.83517i	0.397543	−	0.917584i	$-0.369863\pi$
$654$	0	0
$655$	130890.	0.0119207
$656$	0	0
$657$	−1.61224e7	−1.45719
$658$	0	0
$659$	1.68566e7i	1.51202i	0.654562	+	0.756009i	$0.272853\pi$
$659$	1.68566e7i	1.51202i	−0.654562	+	0.756009i	$0.727147\pi$
$660$	0	0
$661$	1.65602e6i	0.147422i	0.997280	+	0.0737111i	$0.0234843\pi$
$661$	1.65602e6i	0.147422i	−0.997280	+	0.0737111i	$0.976516\pi$
$662$	0	0
$663$	−5.80038e6	−0.512475
$664$	0	0
$665$	−44190.3	−0.00387501
$666$	0	0
$667$	− 7.14369e6i	− 0.621739i
$668$	0	0
$669$	2.37401e7i	2.05077i
$670$	0	0
$671$	6.26364e6	0.537057
$672$	0	0
$673$	1.37081e7	1.16665	0.583324	−	0.812240i	$-0.301752\pi$
$673$	1.37081e7	1.16665	0.583324	+	0.812240i	$0.301752\pi$
$674$	0	0
$675$	5.26523e6i	0.444793i
$676$	0	0
$677$	− 2.21494e6i	− 0.185734i	−0.995679	−	0.0928669i	$-0.970397\pi$
$677$	− 2.21494e6i	− 0.185734i	0.995679	−	0.0928669i	$-0.0296031\pi$
$678$	0	0
$679$	−2.01891e6	−0.168052
$680$	0	0
$681$	−1.31031e7	−1.08269
$682$	0	0
$683$	− 532823.i	− 0.0437050i	−0.999761	−	0.0218525i	$-0.993044\pi$
$683$	− 532823.i	− 0.0437050i	0.999761	−	0.0218525i	$-0.00695643\pi$
$684$	0	0
$685$	7641.93i	0 0.000622267i
$686$	0	0
$687$	−1.12384e7	−0.908476
$688$	0	0
$689$	−1.65060e7	−1.32463
$690$	0	0
$691$	5.63330e6i	0.448816i	0.974495	+	0.224408i	$0.0720448\pi$
$691$	5.63330e6i	0.448816i	−0.974495	+	0.224408i	$0.927955\pi$
$692$	0	0
$693$	− 6.52446e6i	− 0.516074i
$694$	0	0
$695$	−227852.	−0.0178933
$696$	0	0
$697$	−3.37569e6	−0.263197
$698$	0	0
$699$	− 2.15824e7i	− 1.67073i
$700$	0	0
$701$	− 5.94926e6i	− 0.457265i	−0.973513	−	0.228633i	$-0.926575\pi$
$701$	− 5.94926e6i	− 0.457265i	0.973513	−	0.228633i	$-0.0734254\pi$
$702$	0	0
$703$	−734207.	−0.0560312
$704$	0	0
$705$	−205970.	−0.0156074
$706$	0	0
$707$	− 1.04398e7i	− 0.785494i
$708$	0	0
$709$	− 8.89034e6i	− 0.664206i	−0.943243	−	0.332103i	$-0.892242\pi$
$709$	− 8.89034e6i	− 0.664206i	0.943243	−	0.332103i	$-0.107758\pi$
$710$	0	0
$711$	−2.31796e7	−1.71962
$712$	0	0
$713$	5.81358e6	0.428272
$714$	0	0
$715$	135881.i	0.00994015i
$716$	0	0
$717$	8.87308e6i	0.644579i
$718$	0	0
$719$	1.23030e7	0.887539	0.443769	−	0.896141i	$-0.353641\pi$
$719$	1.23030e7	0.887539	0.443769	+	0.896141i	$0.353641\pi$
$720$	0	0
$721$	−1.10943e7	−0.794811
$722$	0	0
$723$	− 1.44516e7i	− 1.02818i
$724$	0	0
$725$	− 1.86986e7i	− 1.32119i
$726$	0	0
$727$	4.81217e6	0.337679	0.168840	−	0.985644i	$-0.445998\pi$
$727$	4.81217e6	0.337679	0.168840	+	0.985644i	$0.445998\pi$
$728$	0	0
$729$	2.11789e7	1.47599
$730$	0	0
$731$	− 3.19304e6i	− 0.221010i
$732$	0	0
$733$	2.11550e6i	0.145430i	0.997353	+	0.0727148i	$0.0231663\pi$
$733$	2.11550e6i	0.145430i	−0.997353	+	0.0727148i	$0.976834\pi$
$734$	0	0
$735$	−294951.	−0.0201387
$736$	0	0
$737$	4.28481e6	0.290578
$738$	0	0
$739$	− 3.53365e6i	− 0.238019i	−0.992893	−	0.119010i	$-0.962028\pi$
$739$	− 3.53365e6i	− 0.238019i	0.992893	−	0.119010i	$-0.0379719\pi$
$740$	0	0
$741$	− 3.56034e6i	− 0.238202i
$742$	0	0
$743$	−2.10515e7	−1.39898	−0.699489	−	0.714643i	$-0.746589\pi$
$743$	−2.10515e7	−1.39898	−0.699489	+	0.714643i	$0.746589\pi$
$744$	0	0
$745$	−586112.	−0.0386892
$746$	0	0
$747$	1.93763e7i	1.27049i
$748$	0	0
$749$	3.01033e7i	1.96070i
$750$	0	0
$751$	9.76794e6	0.631980	0.315990	−	0.948763i	$-0.397663\pi$
$751$	9.76794e6	0.631980	0.315990	+	0.948763i	$0.397663\pi$
$752$	0	0
$753$	−1.09223e7	−0.701983
$754$	0	0
$755$	− 55373.8i	− 0.00353538i
$756$	0	0
$757$	3.90009e6i	0.247363i	0.992322	+	0.123682i	$0.0394701\pi$
$757$	3.90009e6i	0.247363i	−0.992322	+	0.123682i	$0.960530\pi$
$758$	0	0
$759$	−3.63753e6	−0.229194
$760$	0	0
$761$	−1.28261e7	−0.802845	−0.401422	−	0.915893i	$-0.631484\pi$
$761$	−1.28261e7	−0.802845	−0.401422	+	0.915893i	$0.631484\pi$
$762$	0	0
$763$	7.93082e6i	0.493181i
$764$	0	0
$765$	140851.i	0.00870174i
$766$	0	0
$767$	2.54588e7	1.56261
$768$	0	0
$769$	1.15731e7	0.705724	0.352862	−	0.935675i	$-0.385208\pi$
$769$	1.15731e7	0.705724	0.352862	+	0.935675i	$0.385208\pi$
$770$	0	0
$771$	1.37725e7i	0.834404i
$772$	0	0
$773$	− 3.69152e6i	− 0.222206i	−0.993809	−	0.111103i	$-0.964562\pi$
$773$	− 3.69152e6i	− 0.222206i	0.993809	−	0.111103i	$-0.0354384\pi$
$774$	0	0
$775$	1.52171e7	0.910073
$776$	0	0
$777$	−1.40329e7	−0.833861
$778$	0	0
$779$	− 2.07204e6i	− 0.122336i
$780$	0	0
$781$	7.67855e6i	0.450455i
$782$	0	0
$783$	1.00939e7	0.588378
$784$	0	0
$785$	508998.	0.0294810
$786$	0	0
$787$	3.15075e7i	1.81333i	0.421851	+	0.906665i	$0.361381\pi$
$787$	3.15075e7i	1.81333i	−0.421851	+	0.906665i	$0.638619\pi$
$788$	0	0
$789$	− 9.72652e6i	− 0.556244i
$790$	0	0
$791$	−2.74965e7	−1.56256
$792$	0	0
$793$	3.68487e7	2.08084
$794$	0	0
$795$	710599.i	0.0398755i
$796$	0	0
$797$	− 1.13115e7i	− 0.630775i	−0.948963	−	0.315387i	$-0.897866\pi$
$797$	− 1.13115e7i	− 0.630775i	0.948963	−	0.315387i	$-0.102134\pi$
$798$	0	0
$799$	2.03682e6	0.112872
$800$	0	0
$801$	3.35583e7	1.84807
$802$	0	0
$803$	− 6.62156e6i	− 0.362386i
$804$	0	0
$805$	265606.i	0.0144460i
$806$	0	0
$807$	2.70235e7	1.46069
$808$	0	0
$809$	−1.87371e7	−1.00654	−0.503271	−	0.864128i	$-0.667870\pi$
$809$	−1.87371e7	−1.00654	−0.503271	+	0.864128i	$0.667870\pi$
$810$	0	0
$811$	− 1.04019e6i	− 0.0555341i	−0.999614	−	0.0277671i	$-0.991160\pi$
$811$	− 1.04019e6i	− 0.0555341i	0.999614	−	0.0277671i	$-0.00883967\pi$
$812$	0	0
$813$	− 2.87108e7i	− 1.52342i
$814$	0	0
$815$	−22206.2	−0.00117106
$816$	0	0
$817$	1.95992e6	0.102727
$818$	0	0
$819$	− 3.83831e7i	− 1.99954i
$820$	0	0
$821$	2.89521e7i	1.49907i	0.661965	+	0.749535i	$0.269723\pi$
$821$	2.89521e7i	1.49907i	−0.661965	+	0.749535i	$0.730277\pi$
$822$	0	0
$823$	1.73232e7	0.891517	0.445758	−	0.895153i	$-0.352934\pi$
$823$	1.73232e7	0.891517	0.445758	+	0.895153i	$0.352934\pi$
$824$	0	0
$825$	−9.52125e6	−0.487034
$826$	0	0
$827$	− 7.19436e6i	− 0.365787i	−0.983133	−	0.182893i	$-0.941454\pi$
$827$	− 7.19436e6i	− 0.365787i	0.983133	−	0.182893i	$-0.0585463\pi$
$828$	0	0
$829$	2.28187e7i	1.15320i	0.817026	+	0.576600i	$0.195621\pi$
$829$	2.28187e7i	1.15320i	−0.817026	+	0.576600i	$0.804379\pi$
$830$	0	0
$831$	−1.90305e7	−0.955978
$832$	0	0
$833$	2.91675e6	0.145642
$834$	0	0
$835$	− 343680.i	− 0.0170584i
$836$	0	0
$837$	8.21451e6i	0.405292i
$838$	0	0
$839$	−1.81694e7	−0.891121	−0.445560	−	0.895252i	$-0.646996\pi$
$839$	−1.81694e7	−0.891121	−0.445560	+	0.895252i	$0.646996\pi$
$840$	0	0
$841$	−1.53359e7	−0.747684
$842$	0	0
$843$	− 2.82899e7i	− 1.37108i
$844$	0	0
$845$	285061.i	0.0137340i
$846$	0	0
$847$	−2.32019e7	−1.11126
$848$	0	0
$849$	−3.45376e7	−1.64446
$850$	0	0
$851$	4.41295e6i	0.208884i
$852$	0	0
$853$	2.58313e7i	1.21555i	0.794108	+	0.607776i	$0.207938\pi$
$853$	2.58313e7i	1.21555i	−0.794108	+	0.607776i	$0.792062\pi$
$854$	0	0
$855$	−86455.8	−0.00404463
$856$	0	0
$857$	1.83988e7	0.855731	0.427866	−	0.903842i	$-0.359266\pi$
$857$	1.83988e7	0.855731	0.427866	+	0.903842i	$0.359266\pi$
$858$	0	0
$859$	2.69993e7i	1.24845i	0.781246	+	0.624224i	$0.214585\pi$
$859$	2.69993e7i	1.24845i	−0.781246	+	0.624224i	$0.785415\pi$
$860$	0	0
$861$	− 3.96028e7i	− 1.82061i
$862$	0	0
$863$	−3.05160e7	−1.39477	−0.697383	−	0.716699i	$-0.745652\pi$
$863$	−3.05160e7	−1.39477	−0.697383	+	0.716699i	$0.745652\pi$
$864$	0	0
$865$	795240.	0.0361375
$866$	0	0
$867$	3.10527e7i	1.40298i
$868$	0	0
$869$	− 9.51999e6i	− 0.427648i
$870$	0	0
$871$	2.52073e7	1.12585
$872$	0	0
$873$	−3.94988e6	−0.175408
$874$	0	0
$875$	1.39088e6i	0.0614141i
$876$	0	0
$877$	− 2.80238e7i	− 1.23035i	−0.788391	−	0.615174i	$-0.789086\pi$
$877$	− 2.80238e7i	− 1.23035i	0.788391	−	0.615174i	$-0.210914\pi$
$878$	0	0
$879$	−1.77242e7	−0.773738
$880$	0	0
$881$	−8.04971e6	−0.349414	−0.174707	−	0.984620i	$-0.555898\pi$
$881$	−8.04971e6	−0.349414	−0.174707	+	0.984620i	$0.555898\pi$
$882$	0	0
$883$	− 8.61922e6i	− 0.372020i	−0.982548	−	0.186010i	$-0.940444\pi$
$883$	− 8.61922e6i	− 0.372020i	0.982548	−	0.186010i	$-0.0595556\pi$
$884$	0	0
$885$	− 1.09603e6i	− 0.0470395i
$886$	0	0
$887$	−6.77038e6	−0.288937	−0.144469	−	0.989509i	$-0.546147\pi$
$887$	−6.77038e6	−0.288937	−0.144469	+	0.989509i	$0.546147\pi$
$888$	0	0
$889$	−1.45404e7	−0.617054
$890$	0	0
$891$	4.72583e6i	0.199427i
$892$	0	0
$893$	1.25022e6i	0.0524636i
$894$	0	0
$895$	422898.	0.0176473
$896$	0	0
$897$	−2.13994e7	−0.888015
$898$	0	0
$899$	− 2.91725e7i	− 1.20386i
$900$	0	0
$901$	− 7.02706e6i	− 0.288378i
$902$	0	0
$903$	3.74600e7	1.52879
$904$	0	0
$905$	−584117.	−0.0237071
$906$	0	0
$907$	− 3.35760e7i	− 1.35522i	−0.735421	−	0.677611i	$-0.763015\pi$
$907$	− 3.35760e7i	− 1.35522i	0.735421	−	0.677611i	$-0.236985\pi$
$908$	0	0
$909$	− 2.04248e7i	− 0.819877i
$910$	0	0
$911$	3.92094e7	1.56529	0.782645	−	0.622469i	$-0.213870\pi$
$911$	3.92094e7	1.56529	0.782645	+	0.622469i	$0.213870\pi$
$912$	0	0
$913$	−7.95797e6	−0.315955
$914$	0	0
$915$	− 1.58637e6i	− 0.0626400i
$916$	0	0
$917$	− 1.51851e7i	− 0.596340i
$918$	0	0
$919$	−1.65806e7	−0.647609	−0.323804	−	0.946124i	$-0.604962\pi$
$919$	−1.65806e7	−0.647609	−0.323804	+	0.946124i	$0.604962\pi$
$920$	0	0
$921$	4.88136e7	1.89623
$922$	0	0
$923$	4.51725e7i	1.74530i
$924$	0	0
$925$	1.15509e7i	0.443876i
$926$	0	0
$927$	−2.17054e7	−0.829601
$928$	0	0
$929$	3.97202e7	1.50998	0.754992	−	0.655734i	$-0.227641\pi$
$929$	3.97202e7	1.50998	0.754992	+	0.655734i	$0.227641\pi$
$930$	0	0
$931$	1.79033e6i	0.0676955i
$932$	0	0
$933$	7.24565e7i	2.72504i
$934$	0	0
$935$	−57848.2	−0.00216402
$936$	0	0
$937$	2.47490e7	0.920891	0.460446	−	0.887688i	$-0.347690\pi$
$937$	2.47490e7	0.920891	0.460446	+	0.887688i	$0.347690\pi$
$938$	0	0
$939$	2.71858e6i	0.100618i
$940$	0	0
$941$	3.63378e7i	1.33778i	0.743361	+	0.668890i	$0.233230\pi$
$941$	3.63378e7i	1.33778i	−0.743361	+	0.668890i	$0.766770\pi$
$942$	0	0
$943$	−1.24540e7	−0.456067
$944$	0	0
$945$	−375298.	−0.0136709
$946$	0	0
$947$	− 5.78262e6i	− 0.209532i	−0.994497	−	0.104766i	$-0.966591\pi$
$947$	− 5.78262e6i	− 0.209532i	0.994497	−	0.104766i	$-0.0334093\pi$
$948$	0	0
$949$	− 3.89543e7i	− 1.40407i
$950$	0	0
$951$	3.06759e7	1.09988
$952$	0	0
$953$	−4.95180e7	−1.76616	−0.883081	−	0.469220i	$-0.844535\pi$
$953$	−4.95180e7	−1.76616	−0.883081	+	0.469220i	$0.844535\pi$
$954$	0	0
$955$	− 587648.i	− 0.0208501i
$956$	0	0
$957$	1.82531e7i	0.644254i
$958$	0	0
$959$	886573.	0.0311292
$960$	0	0
$961$	−4.88839e6	−0.170749
$962$	0	0
$963$	5.88954e7i	2.04652i
$964$	0	0
$965$	517932.i	0.0179042i
$966$	0	0
$967$	1.73855e7	0.597891	0.298945	−	0.954270i	$-0.403365\pi$
$967$	1.73855e7	0.597891	0.298945	+	0.954270i	$0.403365\pi$
$968$	0	0
$969$	1.51573e6	0.0518577
$970$	0	0
$971$	− 5.82121e7i	− 1.98137i	−0.136186	−	0.990683i	$-0.543484\pi$
$971$	− 5.82121e7i	− 1.98137i	0.136186	−	0.990683i	$-0.456516\pi$
$972$	0	0
$973$	2.64341e7i	0.895121i
$974$	0	0
$975$	−5.60130e7	−1.88702
$976$	0	0
$977$	2.03137e7	0.680853	0.340426	−	0.940271i	$-0.389429\pi$
$977$	2.03137e7	0.680853	0.340426	+	0.940271i	$0.389429\pi$
$978$	0	0
$979$	1.37826e7i	0.459593i
$980$	0	0
$981$	1.55162e7i	0.514769i
$982$	0	0
$983$	5.54003e7	1.82864	0.914321	−	0.404991i	$-0.132725\pi$
$983$	5.54003e7	1.82864	0.914321	+	0.404991i	$0.132725\pi$
$984$	0	0
$985$	−156319.	−0.00513357
$986$	0	0
$987$	2.38954e7i	0.780768i
$988$	0	0
$989$	− 1.17801e7i	− 0.382965i
$990$	0	0
$991$	9.26753e6	0.299764	0.149882	−	0.988704i	$-0.452111\pi$
$991$	9.26753e6	0.299764	0.149882	+	0.988704i	$0.452111\pi$
$992$	0	0
$993$	4.77179e7	1.53571
$994$	0	0
$995$	364224.i	0.0116630i
$996$	0	0
$997$	− 5.53912e7i	− 1.76483i	−0.470471	−	0.882415i	$-0.655916\pi$
$997$	− 5.53912e7i	− 1.76483i	0.470471	−	0.882415i	$-0.344084\pi$
$998$	0	0
$999$	−6.23544e6	−0.197676

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.6.b.a.17.1		4
3.2	odd	2		288.6.d.b.145.3		4
4.3	odd	2		8.6.b.a.5.2	yes	4
5.2	odd	4		800.6.f.a.49.1		8
5.3	odd	4		800.6.f.a.49.8		8
5.4	even	2		800.6.d.a.401.4		4
8.3	odd	2		8.6.b.a.5.1	✓	4
8.5	even	2	inner	32.6.b.a.17.4		4
12.11	even	2		72.6.d.b.37.3		4
16.3	odd	4		256.6.a.k.1.1		4
16.5	even	4		256.6.a.n.1.1		4
16.11	odd	4		256.6.a.k.1.4		4
16.13	even	4		256.6.a.n.1.4		4
20.3	even	4		200.6.f.a.149.6		8
20.7	even	4		200.6.f.a.149.3		8
20.19	odd	2		200.6.d.a.101.3		4
24.5	odd	2		288.6.d.b.145.2		4
24.11	even	2		72.6.d.b.37.4		4
40.3	even	4		200.6.f.a.149.4		8
40.13	odd	4		800.6.f.a.49.2		8
40.19	odd	2		200.6.d.a.101.4		4
40.27	even	4		200.6.f.a.149.5		8
40.29	even	2		800.6.d.a.401.1		4
40.37	odd	4		800.6.f.a.49.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.6.b.a.5.1	✓	4	8.3	odd	2
8.6.b.a.5.2	yes	4	4.3	odd	2
32.6.b.a.17.1		4	1.1	even	1	trivial
32.6.b.a.17.4		4	8.5	even	2	inner
72.6.d.b.37.3		4	12.11	even	2
72.6.d.b.37.4		4	24.11	even	2
200.6.d.a.101.3		4	20.19	odd	2
200.6.d.a.101.4		4	40.19	odd	2
200.6.f.a.149.3		8	20.7	even	4
200.6.f.a.149.4		8	40.3	even	4
200.6.f.a.149.5		8	40.27	even	4
200.6.f.a.149.6		8	20.3	even	4
256.6.a.k.1.1		4	16.3	odd	4
256.6.a.k.1.4		4	16.11	odd	4
256.6.a.n.1.1		4	16.5	even	4
256.6.a.n.1.4		4	16.13	even	4
288.6.d.b.145.2		4	24.5	odd	2
288.6.d.b.145.3		4	3.2	odd	2
800.6.d.a.401.1		4	40.29	even	2
800.6.d.a.401.4		4	5.4	even	2
800.6.f.a.49.1		8	5.2	odd	4
800.6.f.a.49.2		8	40.13	odd	4
800.6.f.a.49.7		8	40.37	odd	4
800.6.f.a.49.8		8	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	32.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-23.6095i q^{3} -1.38521i q^{5} -160.704 q^{7} -314.408 q^{9} +O(q^{10})\)	\(q-23.6095i q^{3} -1.38521i q^{5} -160.704 q^{7} -314.408 q^{9} -129.129i q^{11} -759.659i q^{13} -32.7041 q^{15} +323.408 q^{17} +198.511i q^{19} +3794.14i q^{21} +1193.15 q^{23} +3123.08 q^{25} +1685.91i q^{27} -5987.24i q^{29} +4872.45 q^{31} -3048.67 q^{33} +222.609i q^{35} +3698.56i q^{37} -17935.2 q^{39} -10437.9 q^{41} -9873.11i q^{43} +435.521i q^{45} +6297.98 q^{47} +9018.79 q^{49} -7635.50i q^{51} -21728.1i q^{53} -178.871 q^{55} +4686.75 q^{57} +33513.4i q^{59} +48506.8i q^{61} +50526.7 q^{63} -1052.29 q^{65} +33182.4i q^{67} -28169.7i q^{69} -59464.1 q^{71} +51278.6 q^{73} -73734.4i q^{75} +20751.6i q^{77} +73724.5 q^{79} -36597.7 q^{81} -61628.0i q^{83} -447.988i q^{85} -141356. q^{87} -106735. q^{89} +122080. i q^{91} -115036. i q^{93} +274.980 q^{95} +12562.9 q^{97} +40599.2i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 96 q^{7} - 164 q^{9}+O(q^{10}) \) 4 * q - 96 * q^7 - 164 * q^9	\( 4 q - 96 q^{7} - 164 q^{9} + 416 q^{15} + 200 q^{17} - 2336 q^{23} + 1556 q^{25} + 12928 q^{31} - 2352 q^{33} - 35104 q^{39} - 4568 q^{41} + 54720 q^{47} + 9828 q^{49} - 85472 q^{55} - 2032 q^{57} + 153440 q^{63} - 19520 q^{65} - 206688 q^{71} + 39976 q^{73} + 247872 q^{79} + 29684 q^{81} - 307872 q^{87} - 84632 q^{89} + 259744 q^{95} - 99576 q^{97}+O(q^{100}) \) 4 * q - 96 * q^7 - 164 * q^9 + 416 * q^15 + 200 * q^17 - 2336 * q^23 + 1556 * q^25 + 12928 * q^31 - 2352 * q^33 - 35104 * q^39 - 4568 * q^41 + 54720 * q^47 + 9828 * q^49 - 85472 * q^55 - 2032 * q^57 + 153440 * q^63 - 19520 * q^65 - 206688 * q^71 + 39976 * q^73 + 247872 * q^79 + 29684 * q^81 - 307872 * q^87 - 84632 * q^89 + 259744 * q^95 - 99576 * q^97

Introduction

L-functions

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